Often times the terms irony and coincidence are greatly misused in conversation. It is important to understand exactly what the meaning of these two words is so that we can avoid this misuse. No, Alanis Morrissette's song "Ironic" does not help to differentiate between these two words since the most of the events in her song are not actually ironic.
The definition of irony according to the online Webster dictionary is: incongruity between the actual result of a sequence of events and the normal or expected result.
Keeping the meaning of irony in mind, realize that in a nutshell it means that a situation turns out exactly opposite of what you would expect.
Recognize verbal irony when someone makes a statement like "Nice weather we're having out there" when it is pouring down rain and lightening outside. The opposite of what is said is what is meant.
Recognize an example of situational irony when a man makes his living examining air bags in a factory to make sure that they are in working order, but dies in a car wreck because he failed to check his own airbag.
Recognize dramatic irony when an audience is aware that a situation is a certain way, but the character in that play or movie is unaware of that situation. The character is going to be surprised or upset in the end because the result is going to be the opposite of what the character expects.
Another example of irony would be a lady writing a song called Ironic but there really aren't any ironic examples in her song. Well, a few of her examples may qualify.
The definition of coincidence according to the online Webster dictionary is: the occurrence of events that happen at the same time by accident but seem to have some connection.
An example of coincidence would be seeing a friend in another town you both happen to be visiting at the same time.
Another example of a coincidence would be that you are thinking of your husband and decide to call him. You pick up the phone to call him but he is trying to call you at the same time.